Uniform Rectifiability and Harmonic Measure III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided NTA domains

Let $E\subset {\mathbb{R}}^{n+1}$, $n\ge 2$, be a closed, Ahlfors-David regular set of dimension $n$ satisfying the “Riesz Transform bound” $$\underset{\epsilon >0}{\sup }{\int }_E{\mid {\int }_{\{y\in E:\mathrm{\mid }x-y\mathrm{\mid }>\epsilon \}}\frac{x-y}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+1}}f(y)d{H}^n(y)\mid }^{2}d{H}^n(x)\le C{\int }_E\mathrm{\mid }f{\mathrm{\mid }}^{2}d{H}^n\mathrm{.}$$ Assume further that $E$ is the boundary of a domain $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$ satisfying the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition. Then $E$ is uniformly rectifiable.